if y=λx-3 ,y=μx+1 ,y=x+4 are 3 normals from point p to parabola whose axis is along x axis then 2λ-3μ is equal to
Answers
Answer:
kya h ye smjhe me hi ni a rha h
The value of 2λ - 3μ is - 5/2.
Given: The 3 normals of a parabola from a point are given as
y = λx - 3, y = μx + 1, y = x + 4.
To Find: The value of 2λ - 3μ.
Solution:
- It is known that from a point, 3 normals can be drawn to a parabola. Such a point is known as a co-normal point.
- The sum of the ordinates of the co-normal points is zero.
- The ordinates (y) are given as, y = am
Where y = ordinate, a = constant, m = slope.
Coming to the numerical, the normals are given to us to are,
y = λx - 3, ...(1)
y = μx + 1, ...(2)
y = x + 4. ...(3)
These are not in the form of y = am, so to convert we can,
⇒ y + 3 = λx
⇒ Y = λx ...(4)
⇒ y - 1 = μx
⇒ Y = μx ...(5)
⇒ y - 4 = x
⇒ Y = x ...(6)
According to bullet point 2, we can say that,
∑ Y = 0
⇒ y + 3 + y - 1 + y - 4 = 0
⇒ 3y = 2
⇒ y = 2/3
Putting y = 2/3 in y = x + 4, we get;
y = x + 4
⇒ x = 2/3 - 4 = ( 2 - 12 ) / 3
⇒ x = - 10/3
Putting the respective values of x and y in (1), we get;
y = λx - 3
⇒ 2/3 = λ × ( - 10/3 ) - 3
⇒ λ × ( - 10/3 ) = ( 2 + 9 ) / 3
⇒ λ = - 11 / 10 ...(7)
Putting the respective values of x and y in (2), we get;
y = μx + 1
⇒ 2/3 = μ × ( - 10/3 ) + 1
⇒ μ × ( - 10/3 ) = ( 2 - 3 ) / 3
⇒ μ = 1 / 10 ...(8)
Now, we need to find the value of the expression,
2λ - 3μ
Putting the respective values of λ and μ from (7) and (8) in the expression, we get;
⇒ ( 2 × ( - 11/10 )) - ( 3 × 1/10 )
⇒ - 22/10 - 3/10
⇒ - 25/10
⇒ - 5 / 2
Hence, the value of 2λ - 3μ is - 5/2.
#SPJ2